$\begingroup$Note that $\sqrt{Y^2}$ is just $|Y|$. But perhaps you meant to put the summation $\Sigma$ inside the square root. One would be the Absolute Norm and the other would be the Euclidean Norm$\endgroup$
– Alex CJul 16 '17 at 14:25

Minimizing the $L_1$ norm subject to affine constraints is a convex optimization problem. There are a multitude of approaches to solve this problem, and since it's only in two variables, it's quite trivial to solve. Any general purpose optimization library can probably solve this without issues.

Below is a short list of things you can do. It is not an exhaustive list.